Problem 110 - Precalculus with Limits

Question

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 1.3\overline{8} \)

Step-by-Step Solution

Answer

The rational number representation of the repeating decimal \( 1.3\overline{8} \) is \( x = \frac{275}{198} \).

Step 1: Assign a Variable for the Repeating Decimal

First, the repeating decimal is represented as a variable, for instance, \( x = 1.3\overline{8} \). This will serve as the starting point for working out the solution.

Step 2: Express the Decimal Value

Write \( x \) in another format where we shift the decimal place enough steps to the right to have one full repetition behind the decimal point. Since one of our repeating digits is 8, and it is 2 digits from the decimal point, we shift 2 places which leads to \( 100x = 138.8\overline{8} \).

Step 3: Subtract the Original Decimal

Next by subtracting the original decimal equation from this new equation, we can eliminate the repeating part of the decimal. This results in; \( 99x = 137.5 \).

Step 4: Solve for the Variable

Lastly, solve for \( x \) by dividing both sides by 99. Thus, the rational number representation of the repeating decimal is \( x = \frac{137.5}{99} \).

Step 5: Simplify the Fraction

The final step is to simplify the fraction from step 4. We multiply both numerator and denominator by 2 (to get rid of the decimal in the numerator). The final result is \( x = \frac{275}{198} \). This is the rational number representation of the given repeating decimal.

Key Concepts

Repeating Decimals

Repeating decimals are decimals that continue infinitely without adding any new digits or patterns. A classic example is the repeating part of the decimal in the number \( 1.3\overline{8} \). Here, the number 8 is the one that keeps repeating over and over. To show that a digit (or group of digits) repeats, we often use a bar notation, like \( \overline{8} \), which means the digit 8 repeats endlessly after the decimal point.

• A repeating decimal is always a result of dividing one integer by another, meaning it is a rational number.
• Even when a decimal looks like it continues forever, if it repeats, it can be expressed as a fraction.

Once a decimal is known to be repeating, such as \( 1.3\overline{8} \), our next task is to convert it into a fraction, which brings us to the concept of decimal representation.

Decimal Representation

Decimal representation deals with how we write numbers in a system based on powers of ten. In our case, \( 1.3\overline{8} \) includes a whole number (1), followed by a decimal point and a repeating fraction behind it. This fractional part highlights the concept of base 10 representation where each place represents a power of ten.

• The number \( 1.3\overline{8} \) splits into \( 1 + 0.3\overline{8} \).
• Here, \( 0.3\overline{8} \) shows the repeating decimal aspect.

Understanding this helps in shifting the decimal places to form equations necessary to convert these repeating decimals into simple fractions, as was done by multiplying by powers of 10 in the solution steps.

Fraction Simplification

Once we have the fraction form of a repeating decimal, the simplification process involves reducing it to its simplest form. Simplifying fractions means finding an equivalent fraction where the numerator and denominator share no common divisors other than 1. This always makes it easier to work with and understand numbers.

• Initially, the decimal in the numerator \( 137.5 \) was turned into \( 275 \) by multiplying by 2.
• Thus, the equation becomes \( \frac{275}{198} \).

By recognizing and eliminating factors shared by the numerator and the denominator, we make this fraction more manageable, often simplifying tasks in mathematical operations.

Mathematical Operations

These are the processes that involve basic arithmetic actions like addition, subtraction, multiplication, and division. In dealing with repeating decimals, these operations play vital roles in effectively converting decimals to fractions.

• We first multiply by powers of 10 to align repeating sections for subtraction.
• After aligning, subtraction eliminates the repeating part making expressions like \( 99x = 137.5 \).
• Dividing simplifies the expression to find the value of \( x \).

Each operation has its place and function, underpinning the logic that unfolds the mystery of repeating decimals into understandable rational numbers.